Theorem opprvalg 13565

Description: Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypotheses

Ref	Expression
opprval.1	|- B = ( Base ` R )
opprval.2	|- .x. = ( .r ` R )
opprval.3	|- O = (oppr ` R )

Assertion

Ref	Expression
opprvalg	|- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )

Proof of Theorem opprvalg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprval.3	. 2 |- O = (oppr ` R )
2		df-oppr 13564	. . 3 |- oppr = ( x e. _V |-> ( x sSet <. ( .r ` ndx ) , tpos ( .r ` x ) >. ) )
3		id 19	. . . 4 |- ( x = R -> x = R )
4		fveq2 5554	. . . . . . 7 |- ( x = R -> ( .r ` x ) = ( .r ` R ) )
5		opprval.2	. . . . . . 7 |- .x. = ( .r ` R )
6	4, 5	eqtr4di 2244	. . . . . 6 |- ( x = R -> ( .r ` x ) = .x. )
7	6	tposeqd 6301	. . . . 5 |- ( x = R -> tpos ( .r ` x ) = tpos .x. )
8	7	opeq2d 3811	. . . 4 |- ( x = R -> <. ( .r ` ndx ) , tpos ( .r ` x ) >. = <. ( .r ` ndx ) , tpos .x. >. )
9	3, 8	oveq12d 5936	. . 3 |- ( x = R -> ( x sSet <. ( .r ` ndx ) , tpos ( .r ` x ) >. ) = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )
10		elex 2771	. . 3 |- ( R e. V -> R e. _V )
11		mulrslid 12749	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
12	11	simpri 113	. . . . 5 |- ( .r ` ndx ) e. NN
13	12	a1i 9	. . . 4 |- ( R e. V -> ( .r ` ndx ) e. NN )
14	11	slotex 12645	. . . . . 6 |- ( R e. V -> ( .r ` R ) e. _V )
15	5, 14	eqeltrid 2280	. . . . 5 |- ( R e. V -> .x. e. _V )
16		tposexg 6311	. . . . 5 |- ( .x. e. _V -> tpos .x. e. _V )
17	15, 16	syl 14	. . . 4 |- ( R e. V -> tpos .x. e. _V )
18		setsex 12650	. . . 4 |- ( ( R e. _V /\ ( .r ` ndx ) e. NN /\ tpos .x. e. _V ) -> ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) e. _V )
19	10, 13, 17, 18	syl3anc 1249	. . 3 |- ( R e. V -> ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) e. _V )
20	2, 9, 10, 19	fvmptd3 5651	. 2 |- ( R e. V -> (oppr ` R ) = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )
21	1, 20	eqtrid 2238	1 |- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   _Vcvv 2760   <.cop 3621   ` cfv 5254  (class class class)co 5918  tpos ctpos 6297   NNcn 8982   ndxcnx 12615   sSet csts 12616  Slot cslot 12617   Basecbs 12618   .rcmulr 12696  opp_rcoppr 13563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-tpos 6298  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-sets 12625  df-mulr 12709  df-oppr 13564

This theorem is referenced by:  opprmulfvalg  13566  opprex  13569  opprsllem  13570